# How do you write an equation of the line that passes through the given points (0,0), (4,-20)?

Feb 12, 2017

$\left(y + \textcolor{red}{20}\right) = \textcolor{b l u e}{- 5} \left(x - \textcolor{red}{4}\right)$

Or

$y = - 5 x$

#### Explanation:

The point-slope formula can be used to write an equation for this line. First, we must determine the slope. The slope can be found by using the formula: $m = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

Where $m$ is the slope and ($\textcolor{b l u e}{{x}_{1} , {y}_{1}}$) and ($\textcolor{red}{{x}_{2} , {y}_{2}}$) are the two points on the line.

Substituting the values from the two points in the problem gives:

$m = \frac{\textcolor{red}{- 20} - \textcolor{b l u e}{0}}{\textcolor{red}{4} - \textcolor{b l u e}{0}}$

$m = \frac{- 20}{4} = - 5$

The point-slope formula states: $\left(y - \textcolor{red}{{y}_{1}}\right) = \textcolor{b l u e}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$

Where $\textcolor{b l u e}{m}$ is the slope and $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through.

Substituting the slope we calculated and the second point gives:

$\left(y - \textcolor{red}{- 20}\right) = \textcolor{b l u e}{- 5} \left(x - \textcolor{red}{4}\right)$

$\left(y + \textcolor{red}{20}\right) = \textcolor{b l u e}{- 5} \left(x - \textcolor{red}{4}\right)$

We can also substitute the slope we calculated and the first point giving:

$\left(y - \textcolor{red}{0}\right) = \textcolor{b l u e}{- 5} \left(x - \textcolor{red}{0}\right)$

$y = - 5 x$